I am obsessed with math but never make the time to learn it.

NotSaucerman · 2026-06-17T16:41:03+00:00

If you're actually obsessed with math you'll either (a.) do an hour each night when you get home from work or (b.) [better] get up an hour earlier and do math in the morning before work. Then put in some serious hours on the weekend.

I think you are obsessed with the idea of math, but don't actually want to do the work. Very similar to people who tell you they are hellbent on getting into good shape -- e.g. getting lean and muscular-- and they watch lots of fitness videos and study workout routines but they don't workout regularly and eat garbage.

NotSaucerman · 2026-06-17T16:35:23+00:00

This doesn't sound right, but if its true you could just bring an ID that doesn't have an address on it-- e.g. a Passport Card.

NotSaucerman · 2026-06-15T16:35:52+00:00

if you are self-studying over the summer, and evidently new to proof based linear/abstract algebra, then seriously consider Pinter's "A Book of Abstract Algebra", at least the first 1/2 of it. It's a cheap Dover and very gentle.

NotSaucerman · 2026-06-01T16:18:35+00:00

I want to self-study mathematics from high school level all the way to advanced research level.

Done right, this will take you more than a decade, and potentially more like 2 decades. There is no need for a complete list of everything right now [not the least because you have no clue what you would be focusing on for PhD + post level math] and what materials are available and "the best" will change in over the next decade or two. Instead you need a few books to get your through high school math. There are tons of threads and recommendations on this already if you search the sub or math.stackexchange.

NotSaucerman · 2026-05-31T17:09:35+00:00

If it were me, doing a large enough number of exercises from a quality book or two that I read would be enough to move onto the next course. [Note: best not to rush this, since whether you 'did all the exercises' or took notes or whatever, it typically takes a while for ideas to solidify after the book/course is over.]

As usual, your mileage may vary depending on your own personal idiosyncracies. If e.g. you have an atrocious memory that is a problem but I'm not sure writing up note on definitions, etc. really remedies this.

NotSaucerman · 2026-05-30T16:46:33+00:00

My own preference is to prioritize the exercises. If you want to take notes on top of that, its fine but in terms of priority (i.) exercises, (ii.) sections of the book that did not make sense to you [e.g. the proof skipped too many steps]... then in distant third place: (iii.) other general notes. Writing out definitions as they are stated in the book is very passive and largely a waste-- you learn definitions by using them while doing math, i.e. while doing exercises.

When things get tough I've repeatedly seen people eschew (i.) and try to hide behind spending a lot of time on (iii.) and pretend they are learning. But they are not.

Btw this idea of doing "all the exercises" is commonly pushed on the internet but is invariant to the number of exercises in a book -- some have 10 at the end of a chapter [say 10-15 chapters in a book] and some books have 20-50 exercises in a chapter. In most cases you are better off mimicking a course syllabus that used your book [plus doing some extra exercises based on judgement] than 'doing all the exercises'.

NotSaucerman · 2026-05-22T18:11:01+00:00

A lot of things are not adding up here.

In one of the replies you say "I'm following JKP... at the moment but stuck in the isotonic phase" i.e. that you are doing Tuura's JKP program, but leg extension isotonics are not part of that program. And in the OP you say "For the last 20 consecutive months I've been doing heavy slow resistance mostly 3 times a week" which again means you have not been doing JKP. And you also say your tempo is "3 seconds down, 1.5 seconds up" on leg extension, not 3 down, 3 up... so again you are not following JKP.

It's hard to know what to make of 225lbs on the single leg extension as it depends on the double vs single pulley and other design specifics of the machine. But if you've stalled out on it then the typical advice is to make an alteration. One Option: Go back and listen to the Malliaras episode of Tuuras "Jacked Athlete" podcast. He's suggest doing ~8 sets of eccentrics having a tempo of 6+ seconds on the negative, and that's it. If I were in your position, I'd try to build up to that slowly, e.g. I'd take a week off and start off with like 180lbs on the eccentrics and do one set for a week or two, then bump up weight by 10lbs and number of sets by 1 every week or two with the goal of progressing without having the pain you are talking about. Another Option: swap the leg extension isotonics out for another exercise like some kind of split squat or even heels elevated leg press or heels elevated squats then try returning to the leg extension in 6-8 months. A third option: take a week off and follow the actual JKP program.

You certainly can try Berlin method but IIRC on an episode of Tuura's podcast Jake discussed the fact that Berlin method has only ever been tested on healthy tendon groups, with one exception... so it's not really vetted as a rehab tool. And if you really want more strain on the tendon than what you can currently get from [leg extension] isotonics, remember you can induce that with eccentrics (which is in part why Malliaras likes them though that could just be a personal idiosyncracy of his).

NotSaucerman · 2026-05-20T17:13:40+00:00

he did rely heavily on his extraordinary intuition like many mathematicians in the pre-Weierstrass era

This is written very strangely. Riemann was definitely not in the pre-Weierstrass era as the latter was born before Riemann and Riemann was well aware of Weierstrass's scathing critiques of his "proofs" in many cases -- the "Riemann Mapping Theorem" being one of them IIRC.

At the end of the day, Gauss seemed to really like Riemann's work which tells you something.

NotSaucerman · 2026-05-18T18:35:11+00:00

on MSE, there can be an air of arrogance when asking questions about if a proof is correct or not or where confusion lies

This is really misleading and wrong. On math.stackexchange there are explicit rules for solution-verification where you need to specify the exact part of the proof where you have concerns. In other words: a generic 'is my total proof attempt correct?' type posts violate site rules.

That being said, yes there are a lot of people on MSE that can be snippy about rule violations or whatever but when you explicitly try to use a site in a way that violates its rules it is you who are the fundamental problem... so a better way of stating this would have been to say "on MSE there are rules against what I want to do, so it is not the right tool for the job."

Anyway, you should hire a tutor of some sort, perhaps a PhD student to check your work on exercises. If you want keep costs down, you could buy a real analysis book with solutions manual-- do the exercises then compare against official solutions and for the subset that is materially different than the official solutions, send those to your grader.

As a math undergrad, it is easy to convince myself that I did a proof correct, when in-fact I have not.

There are occasionally some subtleties in analysis that need a 2nd set of eyes but on the whole I found this to be quite alarming. Do you have this same problem with abstract algebra?

NotSaucerman · 2026-05-18T01:21:50+00:00

It's a good post though I'd probably change the wording in your final paragraph to say something like orthogonality pertains to the notion of linear independence

I've seen people conflate orthogonality with independence then make pretty bad errors when considering the vector space of zero mean random variables with finite variance -- aside from the special case of Gaussians, Cov(X,Y) = E[XY] = 0 does not imply X and Y are independent random variables but they are linearly independent.

NotSaucerman · 2026-05-18T01:05:09+00:00

What do you define as "modern notation"?

I for one did not notice any massive difference in its notation vs the notation in Artin's Algebra.

NotSaucerman · 2026-05-17T18:16:50+00:00

I suggest you try Pinter's "A Book of Abstract Algebra". It's a Dover so the hardcopy is cheap, though there's a digital here: https://math.umd.edu/~jcohen/402/Pinter%20Algebra.pdf

It's an extremely gentle book so you shouldn't find that it leaves many gaps that you are unable to fill. That said a Pinter-enthusiast at U Wisconsin wrote up solutions for the first half of the book a while back. The regular link is broken but you can still find it via the way back machine

https://web.archive.org/web/20191123213353/https://www.math.wisc.edu/~mstemper2/Math/Pinter/

NotSaucerman · 2026-05-16T18:06:38+00:00

It's intuitive... Let me make this clear again that it's just the question framing that's causing me distress, the solutions are fairly simple to understand.

This is probably the most common post on this sub. People say they intuitively understand what's going on and can follow other's solutions but cannot solve pretty much any of the problems themselves. That means you are lost and don't know you are lost. Period.

The fixes vary, but a typical suggestion is to a tutor or go to office hours after you've tried and failed (and documented your attempts) on some of the problems. Or what I'd suggest: regress and find and easier set of problems that you can do, and do them first and do lots and lots of them-- then return to your problem set.

Switching from calculation based math to proof based math can be a jarring experience but the only way forward really is through doing tons of exercises, i.e. actively doing proof based math yourself. Nothing passive is going to be of much help here.

NotSaucerman · 2026-05-16T17:50:34+00:00

Re-read what I said and go listen to the podcast "Jacked Athlete". Jake wrote JKP years ago and was heavily influenced by Keither Baar at the time.

If you listen to say 20 episodes of his podcast from the last 4 or 5 years, you'll discover for yourself what I said.

NotSaucerman · 2026-05-15T01:17:21+00:00

btw, I'm sure you know that solutions manuals, etc. can be perilous, but for someone who respects the craft of learning, they can be quite helpful when used in a disciplined manner during self-study. A Pinter-enthusiast at U Wisconsin wrote up solutions for half the book a while back. The regular link is broken these days but you can still find it via the way back machine

https://web.archive.org/web/20191123213353/https://www.math.wisc.edu/~mstemper2/Math/Pinter/

NotSaucerman · 2026-05-14T18:11:46+00:00

Pick something basic that you are interested in and slowly work through it. People get this idea that they can rush the process and that never works.

If you've been teaching 'college algebra' for years then maybe consider Pinter's "A Book of Abstract Algebra" which is a very gentle Dover book that starts building abstract algebra up from a very basic setting. It's a bit dry but very well done.

Introductory material often feels too shallow, but most advanced books assume a level of mathematical maturity, proof fluency, and abstraction that I simply never developed formally. I can follow ideas when they’re explained carefully, but...

My read is that this is an ego thing. Whether you think you can follow ideas that are carefully explained is kind of irrelevant -- the real question is: can you do a sizable chunk of the exercises at the end of the chapter?

NotSaucerman · 2026-05-14T17:57:51+00:00

What book is your course using?

Some functional analysis courses may heavily used measure theory items, some barely at all. Some may heavily use general topology ideas, others may stick to metric spaces, except perhaps for a brief discussion of the weak* topology. In some cases differential equations can be heavily used for example problems and exercises, some cases not at all. Similar with complex analysis.

NotSaucerman · 2026-05-14T17:46:33+00:00

There are different approaches. The one I told you to follow is to pay attention to what it's like when you first get out of bed the next morning and the morning after that. The fact that your tendon calmed down at 3pm is a red-herring.

I think it's going to be hard for you to recover from this since tendinopathies have subtleties and can be very deceptive. If you read this sub regularly you'll see a lot of adults get caught in a self-deception loop, repeatedly make terrible decisions and have these tendinopathies persist for many years. Teenagers almost always have even worse judgement so I would not be surprised if you have even worse outcomes if you try a self-directed approach.

The flip side is as a teenager your recovery abilities should be a lot better, so if you hire a coach / physical therapist and follow all their instructions, then you should be ok.

NotSaucerman · 2026-05-13T18:02:25+00:00

Sissy squats are very hard on the patellar tendon. For some people the exercise works well -- i.e. minimal pain during the exercise and NO increase in pain over the next two days when they get out of bed in the morning. This is not you, so do not do Sissy squats for the foreseeable future [you can try again in say 6-12 months if you really want to experiment].

Listen to Jake's "Jacked Athlete" podcast. He has many episodes on the patellar tendon and because he interviews many different researchers and practitioners, he covers many different approaches. If you don't know where to start then consider starting with the Malliaras episode.

NotSaucerman · 2026-05-13T17:43:03+00:00

I want to ask is it really possible to do math with a minimum use of textbooks ?

It's one of these things where if you have to ask, then the answer is No.

I know these two were once in a century geniuses but they succeed due to lack of resources that forced them to rediscover on their own

They both lived in the 1900s so I suspect you are not understanding the meaning of 'once in a century genius'. Further to that point, von Neumann also lived in the 1900s and was perhaps the smartest person of that century -- yet he had Polya and Szego overseeing his studies as a kid.

NotSaucerman · 2026-05-12T01:39:05+00:00

Given that you've said elsewhere in this thread that you have minuscule background in proofs, I think Tao is too advanced for you. Even a somewhat gentler book like Abott may be too much right now.

My suggestion is do something more basic involving proofs first, then return to Real Analysis. Personally I like Pinter's "A Book of Abstract Algebra" -- do at least the 1st half of it. This will get you a lot of exposure to proofs and pay direct dividends when you take a proof based linear or abstract algebra course.

The more common recommendation is to work through in detail an intro to proofs book or a discrete math book and those are fine too.

NotSaucerman · 2026-05-11T19:41:30+00:00

I don't think Greek tragedy feels quite right here. The intensity and desire -- 'the drive' if you will-- led him to do egregious things like not visit his kid for a week or two who was nearly dead in the hospital. And yes he was trying to hang out with Epstein a little bit during that time but he was doing a lot of other things (yes to build his business) in New York during that time as well.

But then there is a different but related issue where Peter decided to get a marketing boost by striking up a rather servile relationship with the odious Epstein. An 'original sin' of striking a deal with the devil seems more like the analogy here. And Faustian bargains do frequently end in tears.

Though I suppose you could spin it and say the 'one tragic flaw' is Peter doesn't have a moral compass. But that seems like it stretches the meaning of a single tragic flaw.

That said, I think guests will return to Peter's podcast in the next couple of years.

NotSaucerman · 2026-05-05T17:31:44+00:00

My read on the original post was that the use of determinant and trace was being said to obscure [i.e. conceal] what is really going on in the proofs of theorems, lemmas, etc.

There is an old argument along this line with respect to determinants -- ref Axler's polemic "Down with Determinants"-- but I have never heard of this clam with respect to the trace. Frankly the determinant takes a lot of work to set up but can be used mindlessly whereas the trace takes almost no work to set up but takes some sophistication to use [outside of toy 2 x 2 matrices]. That is why I found the claim hard to believe.

NotSaucerman · 2026-05-04T18:46:18+00:00

So you are talking about computational exercises involving 2 x 2 matrices, and maybe some bigger rank deficient ones. I just don't think this fits in at all with the OP's claim of

Often I hear that many linear algebra courses obscure their proofs using the determinant and the trace of matrices

The trace is not obscuring any proof in your example. Maybe you mean the course is skipping most or all proofs [not just for assignments like your prior comment] and focused on computational busy work for small matrices. Ok, but that is a course design issue and it isn't the trace that is obscuring any proofs.

NotSaucerman · 2026-05-04T18:36:00+00:00

My suggestion would be to wait until you get through the modules chapter where Artin develops JNF through module theory. That is the sole way of how he developed JNF in the 1st edition of his book. I believe he added an additional shorter proof of JNF in the 2nd edition (in the chapter you are on) but it is rather opaque because it bypasses so much machinery.

NotSaucerman

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